Robust characterizations of k-wise independence over product spaces and related testing results

A discrete distribution D over Σ₁ ×··· ×Σ_n is called (non-uniform) k -wise independent if for any subset of k indices {i₁,...,i_k} and for any z₁∈Σi1,...,z_k∈Σik, Pr_X~D[Xi1···Xik = z₁···z_k] = Pr_X~D[Xi1 = z₁]···Pr_X~D[Xik = z_k]. We study the problem of testing (non-uniform) k -wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from k -wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only when the underlying domain is {0,1}ⁿ. For the non-uniform case, we give a new characterization of distributions being k -wise independent and further show that such a characterization is robust based on our results for the uniform case. These results greatly generalize those of Alon et al. (STOC'07, pp. 496-505) on uniform k -wise independence over the Boolean cubes to non-uniform k -wise independence over product spaces. Our results yield natural testing algorithms for k -wise independence with time and sample complexity sublinear in terms of the support size of the distribution when k is a constant. The main technical tools employed include discrete Fourier transform and the theory of linear systems of congruences.© 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013